NAG Library Function Document nag_tsa_cross_spectrum_bivar (g13cec)

Transcription

1 g13 Time Series Analysis g13cec 1 Purpose NAG Library Function Document nag_tsa_cross_spectrum_bivar (g13cec) For a bivariate time series, nag_tsa_cross_spectrum_bivar (g13cec) calculates the cross amplitude spectrum and squared coherency, together with lower and upper bounds from the univariate and bivariate (cross) spectra. 2 Specification #include <nag.h> #include <nagg13.h> void nag_tsa_cross_spectrum_bivar (const double xg[], const double yg[], const Complex xyg[], Integer ng, const double stats[], double ca[], double calw[], double caup[], double *t, double sc[], double sclw[], double scup[], NagError *fail) 3 Description Estimates of the cross amplitude spectrum A! ð Þ and squared coherency W! ð Þ are calculated for each frequency! as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A! ð Þ ¼ f xy ð! Þ ¼ cf ð! Þ 2 þ qf ð! Þ 2 and W! ð Þ ¼ jf xy ð! Þj 2 f xx ð! Þf yy ð! Þ where: cf ð! Þ and qf ð! Þ are the co-spectrum and quadrature spectrum estimates between the series, i.e., the real and imaginary parts of the cross spectrum f xy ð! Þ as obtained using nag_tsa_spectrum_bivar_cov (g13ccc) or nag_tsa_spectrum_bivar (g13cdc). f xx ð! Þ and f yy ð! Þ are the univariate spectrum estimates for the two series as obtained using nag_tsa_spectrum_univar_cov (g13cac) or nag_tsa_spectrum_univar (g13cbc). The same type and amount of smoothing should be used for these estimates, and this is specified by the degrees of freedom and bandwidth values which are passed from the calls of nag_tsa_spectrum_univar_cov (g13cac) or nag_tsa_spectrum_univar (g13cbc). Upper and lower 95% confidence limits for the cross amplitude are given approximately by p A! ð Þ 1 1:96= ffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d W! ð Þ 1 þ 1, except that a negative lower limit is reset to 0.0, in which case the approximation is rather poor. You are therefore particularly recommended to compare the coherency estimate W! ð Þ with the critical value T derived from the upper 5% point of the F-distribution on ð2;d 2Þ degrees of freedom: 2F T ¼ d 2 þ 2F where d is the degrees of freedom associated with the univariate spectrum estimates. The value of T is returned by the function. The hypothesis that the series are unrelated at frequency!, i.e., that both the true cross amplitude and coherency are zero, may be rejected at the 5% level if W! ð Þ >T. Tests at two frequencies separated by more than the bandwidth may be taken to be independent. The confidence limits on A! ð Þ are strictly appropriate only at frequencies for which the coherency is significant. The same applies to the confidence limits on W! ð Þ which are however calculated at all pffiffiffiffiffiffiffiffiffiffi frequencies using the approximation that arctanh Wl ðþ is Normal with variance 1=d. g13cec.1

2 g13cec NAG Library Manual 4 References Bloomfield P (1976) Fourier Analysis of Time Series: An Introduction Wiley Jenkins G M and Watts D G (1968) Spectral Analysis and its Applications Holden Day 5 Arguments 1: xg½ngš const double Input On entry: the ng univariate spectral estimates, f xx ð! Þ, for the x series. 2: yg½ngš const double Input On entry: the ng univariate spectral estimates, f yy ð! Þ, for the y series. 3: xyg½ngš const Complex Input On entry: f xy ð! Þ, the ng bivariate spectral estimates for the x and y series. The x series leads the y series. Note: the two univariate and the bivariate spectra must each have been calculated using the same amount of smoothing. The frequency width and the shape of the window and the frequency division of the spectral estimates must be the same. The spectral estimates and statistics must also be unlogged 4: ng Integer Input On entry: the number of spectral estimates in each of the arrays xg, yg and xyg. It is also the number of cross amplitude spectral and squared coherency estimates. Constraint: ng 1. 5: stats½4š const double Input On entry: the 4 associated statistics for the univariate spectral estimates for the x and y series. stats½0š contains the degrees of freedom, stats½1š and stats½2š contain the lower and upper bound multiplying factors respectively and stats½3š contains the bandwidth. Constraints: stats½0š 3:0; 0:0 < stats½1š 1:0; stats½2š 1:0. 6: ca½ngš double Output On exit: the ng cross amplitude spectral estimates ^A ð! Þ at each frequency of!. 7: calw½ngš double Output On exit: the ng lower bounds for the ng cross amplitude spectral estimates. 8: caup½ngš double Output On exit: the ng upper bounds for the ng cross amplitude spectral estimates. 9: t double * Output On exit: the critical value for the significance of the squared coherency, T. 10: sc½ngš double Output On exit: the ng squared coherency estimates, ^W ð! Þ at each frequency!. g13cec.2

3 g13 Time Series Analysis g13cec 11: sclw½ngš double Output On exit: the ng lower bounds for the ng squared coherency estimates. 12: scup½ngš double Output On exit: the ng upper bounds for the ng squared coherency estimates. 13: fail NagError * Input/Output The NAG error argument (see Section 3.6 in the Essential Introduction). 6 Error Indicators and Warnings NE_ALLOC_FAIL Dynamic memory allocation failed. NE_BIVAR_SPECTRAL_ESTIM_ZERO A bivariate spectral estimate is zero. For this frequency the cross amplitude spectrum is set to zero, and the contributions to the impulse response function and its standard error are set to zero. NE_INT_ARG_LT On entry, ng ¼ hvaluei. Constraint: ng 1. NE_INTERNAL_ERROR An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance. NE_REAL_ARG_GT On entry, stats½1š must not be greater than 1.0: stats½1š¼hvaluei. NE_REAL_ARG_LE On entry, stats½1š must not be less than or equal to 0.0: stats½1š ¼hvaluei. NE_REAL_ARG_LT On entry, stats½0š must not be less than 3.0: stats½0š¼hvaluei. On entry, stats½2š must not be less than 1.0: stats½2š¼hvaluei. NE_SQUARED_FREQ_GT_ONE A calculated value of the squared coherency exceeds one. For this frequency the squared coherency is reset to one with the result that the cross amplitude spectrum is zero and the contribution to the impulse response function at this frequency is zero. NE_UNIVAR_SPECTRAL_ESTIM_NEG A bivariate spectral estimate is negative. For this frequency the cross amplitude spectrum is set to zero, and the contributions to the impulse response function and its standard error are set to zero. NE_UNIVAR_SPECTRAL_ESTIM_ZERO A bivariate spectral estimate is zero. g13cec.3

4 g13cec NAG Library Manual For this frequency the cross amplitude spectrum is set to zero, and the contributions to the impulse response function and its standard error are set to zero. 7 Accuracy All computations are very stable and yield good accuracy. 8 Further Comments The time taken by nag_tsa_cross_spectrum_bivar (g13cec) is approximately proportional to ng. 9 Example The example program reads the set of univariate spectrum statistics, the 2 univariate spectra and the cross spectrum at a frequency division of 2 20 for a pair of time series. It calls nag_tsa_cross_spectrum_bivar (g13cec) to calculate the cross amplitude spectrum and squared coherency and their bounds and prints the results. 9.1 Program Text /* nag_tsa_cross_spectrum_bivar (g13cec) Example Program. * * Copyright 1996 Numerical Algorithms Group. * * Mark 4, * Mark 8 revised, * #include <nag.h> #include <nagx04.h> #include <stdio.h> #include <nag_stdlib.h> #include <naga02.h> #include <nagg13.h> #define L 80 #define KC 8*L #define NGMAX KC #define NXYMAX 300 int main(int argc, char *argv[]) FILE *fpin, *fpout; Complex *xyg; Integer exit_status = 0, i, is, j, kc = KC, l = L, mw, ng, nxy; NagError fail; double *ca = 0, *calw = 0, *caup = 0, pw, pxy, *sc = 0, *sclw = 0; double *scup = 0, stats[4], t, *x = 0, *xg, *y = 0, *yg; INIT_FAIL(fail); /* Check for command-line IO options fpin = nag_example_file_io(argc, argv, "-data", NULL); fpout = nag_example_file_io(argc, argv, "-results", NULL); fprintf(fpout, "nag_tsa_cross_spectrum_bivar (g13cec) Example Program Results\n"); /* Skip heading in data file fscanf(fpin, "%*[^\n] "); fscanf(fpin, "%ld ", &nxy); if (nxy > 0 && nxy <= NXYMAX) if (!(x = NAG_ALLOC(KC, double)) g13cec.4

5 g13 Time Series Analysis g13cec!(y = NAG_ALLOC(KC, double))!(ca = NAG_ALLOC(NGMAX, double))!(calw = NAG_ALLOC(NGMAX, double))!(caup = NAG_ALLOC(NGMAX, double))!(sc = NAG_ALLOC(NGMAX, double))!(sclw = NAG_ALLOC(NGMAX, double))!(scup = NAG_ALLOC(NGMAX, double))) fprintf(fpout, "Allocation failure\n"); exit_status = -1; for (i = 1; i <= nxy; ++i) fscanf(fpin, "%lf ", &x[i - 1]); for (i = 1; i <= nxy; ++i) fscanf(fpin, "%lf ", &y[i - 1]); /* Set parameters for call to nag_tsa_spectrum_univar (g13cbc) and g13cdc * with mean correction and 10 percent taper pxy = 0.1; /* Window shape parameter and zero covariance at lag 16 pw = 0.5; mw = 16; /* Alignment shift of 3 is = 3; /* Obtain univariate spectrum for the x and the y series /* nag_tsa_spectrum_univar (g13cbc). * Univariate time series, smoothed sample spectrum using * spectral smoothing by the trapezium frequency (Daniell) * window nag_tsa_spectrum_univar(nxy, Nag_Mean, pxy, mw, pw, l, kc, Nag_Unlogged, x, &xg, &ng, stats, &fail); if (fail.code!= NE_NOERROR) fprintf(fpout, "Error from nag_tsa_spectrum_univar (g13cbc).\n%s\n", fail.message); exit_status = 1; /* nag_tsa_spectrum_univar (g13cbc), see above. nag_tsa_spectrum_univar(nxy, Nag_Mean, pxy, mw, pw, l, kc, Nag_Unlogged, y, &yg, &ng, stats, &fail); if (fail.code!= NE_NOERROR) fprintf(fpout, "Error from nag_tsa_spectrum_univar (g13cbc).\n%s\n", fail.message); exit_status = 1; /* Obtain cross spectrum of the bivariate series /* nag_tsa_spectrum_bivar (g13cdc). * Multivariate time series, smoothed sample cross spectrum * using spectral smoothing by the trapezium frequency * (Daniell) window nag_tsa_spectrum_bivar(nxy, Nag_Mean, pxy, mw, is, pw, l, kc, x, y, &xyg, &ng, &fail); if (fail.code!= NE_NOERROR) fprintf(fpout, "Error from nag_tsa_spectrum_bivar (g13cdc).\n%s\n", fail.message); exit_status = 1; /* nag_tsa_cross_spectrum_bivar (g13cec). * Multivariate time series, cross amplitude spectrum, g13cec.5

6 g13cec NAG Library Manual * squared coherency, bounds, univariate and bivariate * (cross) spectra nag_tsa_cross_spectrum_bivar(xg, yg, xyg, ng, stats, ca, calw, caup, &t, sc, sclw, scup, &fail); if (fail.code!= NE_NOERROR) fprintf(fpout, "Error from nag_tsa_cross_spectrum_bivar (g13cec).\n%s\n", fail.message); exit_status = 1; fprintf(fpout, "\n"); fprintf(fpout, " Cross amplitude spectrum\n\n"); fprintf(fpout, " Lower Upper\n"); fprintf(fpout, " Value bound bound\n\n"); for (j = 1; j <= ng; ++j) fprintf(fpout, " %5ld%10.4f%10.4f%10.4f\n", j - 1, ca[j - 1], calw[j - 1], caup[j - 1]); fprintf(fpout, "\n"); fprintf(fpout, " Squared coherency test statistic =%12.4f\n\n", t); fprintf(fpout, " Squared coherency\n\n"); fprintf(fpout, " Lower Upper\n"); fprintf(fpout, " Value bound bound\n\n"); for (j = 1; j <= ng; ++j) fprintf(fpout, " %5ld%10.4f%10.4f%10.4f\n", j - 1, sc[j - 1], sclw[j - 1], scup[j - 1]); NAG_FREE(xg); NAG_FREE(yg); NAG_FREE(xyg); END: if (fpin!= stdin) fclose(fpin); if (fpout!= stdout) fclose(fpout); if (x) NAG_FREE(x); if (y) NAG_FREE(y); if (ca) NAG_FREE(ca); if (calw) NAG_FREE(calw); if (caup) NAG_FREE(caup); if (sc) NAG_FREE(sc); if (sclw) NAG_FREE(sclw); if (scup) NAG_FREE(scup); return exit_status; 9.2 Program Data nag_tsa_cross_spectrum_bivar (g13cec) Example Program Data g13cec.6

7 g13 Time Series Analysis g13cec Program Results nag_tsa_cross_spectrum_bivar (g13cec) Example Program Results Cross amplitude spectrum Lower Upper Value bound bound g13cec.7

8 g13cec NAG Library Manual Squared coherency test statistic = Squared coherency Lower Upper Value bound bound g13cec.8 (last)